Solving Limit Questions: Expert Help with 3^x+4^x/7^x at Infinity

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Homework Statement



Lim (3^x+4^x)/(7^x) x -> ∞

Homework Equations





The Attempt at a Solution



I broke the limit up into two parts:

Lim (3^x)/(7^x) x -> ∞ + Lim (4^x)/(7^x) x -> ∞

However, I don't know where to go from here.
 
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student93 said:

Homework Statement



Lim (3^x+4^x)/(7^x) x -> ∞

Homework Equations





The Attempt at a Solution



I broke the limit up into two parts:

Lim (3^x)/(7^x) x -> ∞ + Lim (4^x)/(7^x) x -> ∞

However, I don't know where to go from here.

Note that ##\frac{3^x}{7^x} = (\frac{3}{7})^x##.

What's the limit of ##a^x## where ##0 < a < 1## as x goes to infinity?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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